B. Jacobs. Categorical Logic and Type Theory. Elsevier Science, 1999.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....f ; t C;B in C , and f C = s A;C ; id C f ; t C;TB ; Ts C;B . It is easy to check that for f : A A we have A g; f B g;y A ;B f B;A B= 0 ;B 0 (in C ) so if T is commutative, they are equal, and define a symmetric monoidal structure on C T . 2. 3 Fibrations Fibrations [Jac99, Bor94b] provide a categorical framework for indexing and substitution. They have been used in Computer Science, for example, to model type theories [Jac99, Cro93] such as polymorphic and dependent types, and to model renaming in models of concurrency [WN95] In this thesis, we will use the dual concept ....

.... B= 0 ;B 0 (in C ) so if T is commutative, they are equal, and define a symmetric monoidal structure on C T . 2.3 Fibrations Fibrations [Jac99, Bor94b] provide a categorical framework for indexing and substitution. They have been used in Computer Science, for example, to model type theories [Jac99, Cro93] such as polymorphic and dependent types, and to model renaming in models of concurrency [WN95] In this thesis, we will use the dual concept of co fibrations to model substitution, so we will only define co fibrations here. The two concepts are dual in the following sense: A functor p : C B is ....

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Bart Jacobs. Categorical Logic and Type Theory. Elsevier, 1999.

.... of classes axiomatized in [19] As usual, an internal category, K, in C is given by an object (i.e. a class) jKj, of K objects, and an internal family, fK(A; B)g A;B : jKj , of K morphisms indexed by domain and codomain, satisfying the expected axioms for identities and composition, see e.g. [17]. We say that an internal category K in C is locally small if the internal family fK(A; B)g A;B : jKj jKj jKj is a small map in C. It is small if, in addition, jKj is small. As expected, small implies locally small, but not vice versa. As the de nitions above suggest, class structure ....

....K is not required to be small. An internal functor, F , from an internal category K to another L is given by a morphism F : jKj jLj; expressing the action on objects, together with a family of maps fF A;B : K(A; B) L(FA; FB)g A;B : jKj that preserve identities and composition, again see [17]. We brie y exhibit S as an internal category in C, before turning attention to P and pP, which are the categories of interest to us. The internal category S is de ned by jSj = P S U S(A; B) B where the family fB g A;B : PSU is formally de ned as an exponential of small objects in the ....

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B. Jacobs. Categorical Logic and Type Theory. North Holland, Amsterdam, 1999.

....cubical sites can be interpreted as classifying categories for these theories. This allows to recover the universal property of the cubical sites and to exhibit them as presentation free versions of the theories. The exposition is modelled on the case of algebraic theories in cartesian categories [9, 20, 25] with the necessary generalisations; some of the ideas behind the analysis can be found in [2] and [21] For conciseness, we restrict here to the framework needed to discuss the cubical sites. Thus, the signatures are single sorted and the languages only allow weakening and exchange as structural ....

B. Jacobs, Categorical logic and type theory, North Holland 1999.

....Our purpose now is to explain how that theory can be extended to include coproducts, whose presence introduces considerable complexity associated with the pariiality of certain path functions expressing the dynamics of the transition structure c. The approach we take is to use type theory [17] to describe the construc tion of sets as types from some base types by forming products, powers and coproducts, and to provide rules of syntax for terms that take values in these types. The base types denote fixed sets of observable elements. There is also the type St of states: this symbol St ....

Bart Jacobs. Categorical Logic and Type Theory. Elsevier, 1999.

....S, P and pP all live as internal categories within C. As usual, an internal category, K, in C is given by an object (i.e. a class) of K objects, and an internal family, of K morphisms indexed by domain and codomain, satisfying the expected axioms for identities and composition, see e.g. [13]. We say that an internal category K in C is locally small if the internal family K K is a small map in C. It is small if, in addition, is small. An internal functor, F , from an internal category K to another L is given by a morphism F : L , expressing the action on objects, ....

....It is small if, in addition, is small. An internal functor, F , from an internal category K to another L is given by a morphism F : L , expressing the action on objects, together with a family FA,B : K(A, B) L(FA, FB) A,B : that preserves identities and composition, again see [13]. We briefly exhibit S as an internal category in C, before turning attention to P and pP, which are the categories of interest to us. The internal category S is defined by S S(A, B) B where the family B A,B is defined as an exponential of small objects in the slice category ....

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B. Jacobs. Categorical Logic and Type Theory. North Holland, Amsterdam, 1999.

....Our purpose now is to explain how that theory can be extended to include coproducts, whose presence introduces considerable complexity associated with the partiality of certain path functions expressing the dynamics of the transition structure . The approach we take is to use type theory [17] to describe the construction of sets as types from some base types by forming products, powers and coproducts, and to provide rules of syntax for terms that take values in these types. The base types denote xed sets of observable elements. There is also the type St of states: this symbol St ....

Bart Jacobs. Categorical Logic and Type Theory. Elsevier, 1999.

....the theory of [4] can be extended to the case of polynomial functors. The presence of coproducts introduces considerable complexity, associated with the partiality of certain path functions that express the dynamics of the transition structure . The approach taken here is to use type theory [8] to describe the construction of sets as types from some base types by forming products, powers and coproducts, and to provide rules of syntax for terms that take values in these types. Among the base types is the type St of states: this symbol St denotes the state set of a given coalgebra. The ....

Jacobs, B., \Categorical Logic and Type Theory," Elsevier, 1999.

.... modest but see the further comments on terminology below. Readers familiar with categories of realizability models based on PCAs will immediately note the similarity of the above de nitions to the well known de nitions of the categories of modest sets and assemblies over a PCA (see, e.g. [19,11,28,26]) Those categories both embed into the so called realizability topos over the PCA [19] We do not get a corresponding embedding into a topos, however; we shall discuss why below. One useful intuition is to think of the category of algebraic lattices as providing a typed universe of realizers ....

....Mod(ALat) and thus PEqu, models dependent type theory. Types are indexed objects of Mod(ALat) the indexing is by objects of Mod(ALat) The regular subobjects can be used to give us logic to reason about the types and with respect to which we have full subset types and full quotient types. See [18,24,26] for more on subset types and quotient types. The same holds for Assm(ALat) but here, in addition, the logic is higher order in short, the point is that the regular subobject classi er is not an object of Mod(ALat) but it is an object of Assm(ALat) we explain this in more detail below. All ....

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B. Jacobs. Categorical Logic and Type Theory. Elsevier Science, 1999.

....propositional level, bi cartesian closed categories, 3 such as presheaf categories of the form Set W , where W is a small category of worlds, are sucient. At the predicate level, the essential structure of proofs is captured by brations, or indexed categories, with speci ed extra structure [17,25,6], q.v. Figure 2: The arrows in the base category are substitutions, i.e. maps between the sets Y and X of variables; The arrows in the bre F (X) over the set X of variables interpret proofs of sequents formed using the variables in X; Substitution lifts from the base to the ....

Jacobs, B., \Categorical Logic and Type Theory", Elsevier, 1998.

....a Martin L of style dependent type theory [ML84, ML98] For the formulation of bracket types we do not need dependent sums or products, but we sometimes assume that they are present in the type theory. We work in a type theory with strong and extensional equality and strong dependent sums, cf. [Jac99]. For reference, we list the rules in Appendix A. Among the types, there are some that satisfy the following condition of proof irrelevance : P type q : P p : P p = q : P (1) In words, this means that any two terms p and q of such a type P are (extensionally) equal. We call the types ....

....: B C and f : A B is only isomorphic to the pullback along 9 the composition g f , whereas for a completely water tight interpretation equality is required. There are several standard ways of resolving this problem, most notably by interpreting the type theory in a suitable bered category [Jac99], and then applying technical results pertaining to these [Hof95] We do not wish to obscure matters by employing such technical devices. The interested reader may either translate our presentation into a suitable bered setting, or assume some other remedy, such as making a coherent choice of ....

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B. Jacobs. Categorical Logic and Type Theory. Elsevier Science, 1999.

....Preliminaries 25 It is easy to check that for f : A A 0 and g : B B 0 we have A g; f B 0 = f g;y A ;B 0 and f B;A 0 B= f g; y A 0 ;B 0 (in C ) so if T is commutative, they are equal, and define a symmetric monoidal structure on C T . 2. 3 Fibrations Fibrations [Jac99, Bor94b] provide a categorical framework for indexing and substitution. They have been used in Computer Science, for example, to model type theories [Jac99, Cro93] such as polymorphic and dependent types, and to model renaming in models of concurrency [WN95] In this thesis, we will use the dual concept ....

....g; y A 0 ;B 0 (in C ) so if T is commutative, they are equal, and define a symmetric monoidal structure on C T . 2.3 Fibrations Fibrations [Jac99, Bor94b] provide a categorical framework for indexing and substitution. They have been used in Computer Science, for example, to model type theories [Jac99, Cro93] such as polymorphic and dependent types, and to model renaming in models of concurrency [WN95] In this thesis, we will use the dual concept of co fibrations to model substitution, so we will only define co fibrations here. The two concepts are dual in the following sense: A functor p : C B is ....

[Article contains additional citation context not shown here]

Bart Jacobs. Categorical Logic and Type Theory. Elsevier, 1999.

....with a summary of notions related to internal categories in C. As usual, an internal category, C, is given by an object C of C objects, and an internal family C(A, B) A,B# C of C morphisms indexed by domain and codomain, satisfying the familiar axioms for identities and composition [11]. Importantly, the presence of class structure also allows notions of size to be dealt with internally in C. For example, we say that an internal category C is locally small if the map C(A, B) A,B# C #dom ,cod# # C C is small. Similarly, we say that C is small if it is locally ....

....with class structure. In this section we briefly review the notions we shall require from the theory of fibrations, 5 and we also give several definitions and results specific to the case in which the base category of the fibration has class structure. Our main reference on fibrations is [11]. We shall assume, without further mention, that all fibrations are cloven, thus we have chosen reindexing functors between fibre categories. Recall the notion of a fibration p : F # C over an arbitrary category C. For the most part, we follow standard terminology, as in [11] We make one ....

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B. Jacobs. Categorical Logic and Type Theory. North Holland, Amsterdam, 1999.

....of . If y(A; A y (A) is a particular choice of cocartesian liftings of : L M 2 L for every A over L, the assignment A 7 y (A) extends to a functor y : E L E M . A functor obtained in this way is called a co reindexing. We will use the bred terminology freely and refer to [4, 9] regarding further reading on this subject. That the cobration associated to a parameterised signature is indeed a cobration follows from ( id C ) being a cocartesian lifting of : L M 2 L for every C over L. We conclude this section with some properties of cobrations of coalgebras. ....

....That f is cartesian provides us with a unique h 0 : UA X with g 0 = f ffi h 0 and by adjunction we obtain the required h : A FX. 3) 1 is a bibration and p a cobration. For p being a bibration, it is suOEcient to show that co reindexing functors y have right adjoints (see [9], 9.1.2) Consider the diagram below. E L y UL A A A A A A A A A E M UM C 2 F is bred (dualise for U cobred ) ioe pF = q and F preserves cartesian liftings. Note that this denition makes sense even when p fails to be a bration. 3 Note that this need neither be ....

Bart Jacobs. Categorical Logic and Type Theory. Elsevier, 1998.

....(via the powerset functor) This paper applies the semantical approach from [12] to the many sorted modal logic from [22] This involves the move from single sorted to many sorted BAOs, via the introduction of appropriately indexed BAOs. This follows general ideas in categorical logic (see [10]) where, for example, models of many sorted predicate logic are described as Boolean algebras indexed by the sorts. Technically, this indexing takes the form of an indexed category or, alternatively, of a bration. Here we shall use the slightly more elementary notion of indexed category, which, ....

B. Jacobs. Categorical Logic and Type Theory. North Holland, Amsterdam, 1999.

....a fibred approach to Birkho#. This only serves as illustration, and will be elaborated in a later publication. 2 Fibrations induced by factorization systems We begin with a review of factorization systems and fibrations. For more details on the former, see [AHS90] or [Bor94] and for the latter, [Jac99]. Definition 2.1 Let H, S be subclasses of the category B # of arrows in an arbitrary category B. We say that S# is a factorization system for B if the following hold. Iso H S (the first of many abuses of set notation for classes) H and S are closed under composition; H and S ....

....morphism above u with domain X. We have the following characterization of uX: For any object Y and arrow f : X ## Y in E such that pf factors through u, say via v, there is a unique h : X ## Y over v such that h uX = f . pY pf ## u ## B # # # # Recall Lemma 9.1. 2 in [Jac99]. Lemma 2.6 A fibration is a bifibration just in case each substitution functor u # :EB EA has a left adjoint, denoted . Remark 2.7 A bifibration p :E ## B is said to satisfy Beck Chevalley (for coproducts) just in case, for every pullback square in B, the canonical ....

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Bart Jacobs. Categorical Logic and Type Theory. Elsevier, 1999.

....of a bred approach to Birkho . This only serves as illustration, and will be elaborated in a later publication. 2 Fibrations induced by factorization systems We begin with a review of factorization systems and brations. For more details on the former, see [AHS90] or [Bor94] and for the latter, [Jac99]. De nition 2.1 Let H, S be subclasses of the category B of arrows in an arbitrary category B . We say that hH; Si is a factorization system for B if the following hold. Iso H S (the rst of many abuses of set notation for classes) H and S are closed under composition; H and S ....

....morphism above u with domain X. We have the following characterization of uX: For any object Y and arrow f : X ## Y in E such that pf factors through u, say via v, there is a unique h : u X ## Y over v such that h uX = f . pY pf ## u ## B # # # # Recall Lemma 9.1. 2 in [Jac99]. Lemma 2.6 A bration is a bi bration just in case each substitution functor u :E B E A has a left adjoint, denoted u : E A ## E B . Remark 2.7 A bi bration p :E ## B is said to satisfy Beck Chevalley (for coproducts) just in case, for every pullback square in B , the ....

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Bart Jacobs. Categorical Logic and Type Theory. Elsevier, 1999.

....is an essential ingredient for the existence of limits. This comprehension can be understood in terms of greatest subcoalgebras , induced by predicates. The second topic of this paper investigates this aspect of comprehension in the framework of fibrations (also called fibred categories) [10]. It is shown that these greatest coalgebras typically come about via greatest invariants. The latter form a notion which can be expressed appropriately in the logic of a fibration via so called predicate lifting [6] The contributions of this paper are five fold. It introduces a notion of ....

....We write Pred for the category of predicates (P X) on sets. Its morphisms (P X) Q Y ) are functions f : X Y mapping P to Q: f(x) 2 Q for all x 2 P . There is then an obvious forgetful functor C: Pred Sets mapping a predicate (P X) to its carrier X . It is a fibration , see [10], but that is not very relevant at this stage. There is a truth functor : Sets Pred sending a set X to the truth predicate (X) X X) It is not hard to see that is right adjoint to C. Additionally, there is a comprehension or subset type functor fg: Pred Sets sending predicates ....

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B. Jacobs. Categorical Logic and Type Theory. North Holland, Amsterdam, 1999.

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B. Jacobs. Categorical Logic and Type Theory. Elsevier Science, 1999.

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B. Jacobs. Categorical Logic and Type Theory. Elsevier Science, 1999.

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B. Jacobs. Categorical Logic and Type Theory. Elsevier, 1999.

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B. Jacobs. Categorical Logic and Type Theory. Elsevier, 1999.

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B. Jacobs. Categorical Logic and Type Theory. Elsevier Science, 1999.

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B. Jacobs. Categorical Logic and Type Theory. Elsevier, 1999.

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Bart Jacobs. Categorical Logic and Type Theory. Elsevier, 1999. Robert Goldblatt

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Bart Jacobs. Categorical Logic and Type Theory. Elsevier, 1999.

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